Counting faces of cubical spheres modulo 2
Eric Babson and Clara Chan

Several recent papers have addressed the problem of characterizing the f-vectors (numbers of faces of each dimension) of cubical polytopes. Along these lines Blind and Blind have shown that unlike in the simplicial case, there are parity restrictions on the f-vectors of cubical polytopes. In particular, except for polygons, all even dimensional cubical polytopes have an even number of vertices. We show that this holds for all odd-dimensional cubical spheres, and that any other modular equations which hold for the f-vectors of all d-dimensional cubical spheres must also be modulo 2. We then reduce the question of which mod 2 equations hold for f-vectors of cubical d-spheres to one about the Euler characteristics of multiple point loci from codimension 1 immersions into the d-sphere. Some results about this topological question are known (e.g., Eccles, Herbert) and translate immediately to modulo 2 results about cubical spheres.